Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An application of Macaulay's estimate to sums of squares problems in several complex variables

Authors: Dusty Grundmeier and Jennifer Halfpap Kacmarcik
Journal: Proc. Amer. Math. Soc. 143 (2015), 1411-1422
MSC (2010): Primary 13D40, 32A17, 32H99
Published electronically: December 9, 2014
MathSciNet review: 3314056
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials $ r(z,\bar {z})$ on $ \mathbb{C}^n \times \mathbb{C}^n$ for which $ r(z,\bar {z})\left \lVert z \right \rVert ^{2d}=\left \lVert h(z) \right \rVert ^2$ for some natural number $ d$ and a holomorphic polynomial mapping $ h=(h_1, \ldots , h_K)$ from $ \mathbb{C}^n$ to $ \mathbb{C}^K$. When $ r$ has this property for some $ d$, one seeks relationships between $ d$, $ K$, and the signature and rank of the coefficient matrix of $ r$. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in $ \mathbb{C}[z_1,\ldots ,z_n]$ and apply a well-known result of Macaulay to estimate some natural quantities.

References [Enhancements On Off] (What's this?)

  • [CD96] David W. Catlin and John P. D'Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, Math. Res. Lett. 3 (1996), no. 2, 149-166. MR 1386836 (97f:32025),
  • [CD97] David W. Catlin and John P. D'Angelo, Positivity conditions for bihomogeneous polynomials, Math. Res. Lett. 4 (1997), no. 4, 555-567. MR 1470426 (98e:32023),
  • [CD99] David W. Catlin and John P. D'Angelo, An isometric imbedding theorem for holomorphic bundles, Math. Res. Lett. 6 (1999), no. 1, 43-60. MR 1682713 (2000g:32023),
  • [D'A93] John P. D'Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR 1224231 (94i:32022)
  • [D'A02] John P. D'Angelo.
    Inequlaities from complex analysis.
    Carus Mathematical Monographs. MAA, 2002.
  • [D'A05] John P. D'Angelo, Complex variables analogues of Hilbert's seventeenth problem, Internat. J. Math. 16 (2005), no. 6, 609-627. MR 2153486 (2006h:32004),
  • [D'A11] John P. D'Angelo, Hermitian analogues of Hilbert's 17-th problem, Adv. Math. 226 (2011), no. 5, 4607-4637. MR 2770459 (2012a:12001),
  • [DZ13] Alexis Drouot and Maciej Zworski, A quantitative version of Catlin-D'Angelo-Quillen theorem, Anal. Math. Phys. 3 (2013), no. 1, 1-19. MR 3015627,
  • [Ebe13] Peter Ebenfelt, Partial rigidity of degenerate CR embeddings into spheres, Adv. Math. 239 (2013), 72-96. MR 3045142,
  • [EHZ05] Peter Ebenfelt, Xiaojun Huang, and Dmitri Zaitsev, The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics, Amer. J. Math. 127 (2005), no. 1, 169-191. MR 2115664 (2006c:32047)
  • [GLV11] Dusty Grundmeier, Jiří Lebl, and Liz Vivas.
    Bounding the rank of hermitian forms and rigidity for CR mappings of hyperquadrics.
    arxiv:1110.4082v2[mathCV], 2011.
  • [Gre89] Mark Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 76-86. MR 1023391 (90k:13021),
  • [Gre10] Mark L. Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119-186. MR 1648665 (99m:13040)
  • [Gru11] Dusty Grundmeier, Signature pairs for group-invariant Hermitian polynomials, Internat. J. Math. 22 (2011), no. 3, 311-343. MR 2782691 (2012c:32031),
  • [HJY13] Xiaojun Huang, Shanyu Ji, and Wanke Yin,
    On the third gap for proper holomorphic maps between balls.
    Mathematische Annalen, pages 1-28, 2013.
  • [HL13] Jennifer Halfpap and Jiří Lebl.
    Signature pairs of positive polynomials.
    Bull. Inst. Math. Acad. Sin. (N.S.), 8(2):169-192, 2013.
  • [Hua99] Xiaojun Huang, On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions, J. Differential Geom. 51 (1999), no. 1, 13-33. MR 1703603 (2000e:32020)
  • [Mac27] F. S. Macaulay,
    Some properties of enumeration in the theory of modular systems.
    Proc. Lond. Math. Soc., 26:531-555, 1927.
  • [Qui68] Daniel G. Quillen, On the representation of hermitian forms as sums of squares, Invent. Math. 5 (1968), 237-242. MR 0233770 (38 #2091)
  • [TY06] Wing-Keung To and Sai-Kee Yeung, Effective isometric embeddings for certain Hermitian holomorphic line bundles, J. London Math. Soc. (2) 73 (2006), no. 3, 607-624. MR 2241969 (2008e:32027),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13D40, 32A17, 32H99

Retrieve articles in all journals with MSC (2010): 13D40, 32A17, 32H99

Additional Information

Dusty Grundmeier
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306

Jennifer Halfpap Kacmarcik
Affiliation: Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812

Received by editor(s): March 31, 2013
Received by editor(s) in revised form: August 2, 2013
Published electronically: December 9, 2014
Additional Notes: The first author was partially supported by NSF RTG grant DMS-1045119.
The second author was supported in part by NSF grant DMS 1200815.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society